Differentiate the logarithmic function

Question

y=log₂(xlog₅x)

For this one, should I try to change the base to lnx? Or should I just proceed and differentiate the function + chain rule everything?

So like, 1/[(xlog5x)ln2]*d/dx[xlog5x]?

Answer 1

y = log₂(xlog₅x) = log₂x + log₂(log₅x) = lnx / ln2 + ln(log₅x) / ln2

= (1/ln2) [ lnx + ln(log₅x) ] = (1/ln2) [ lnx + ln(lnx / ln5) ] = (1/ln2) [ lnx + ln(lnx) - ln(ln5) ]

So, y' = (1/ln2) [ 1/x + (1/x) / lnx - 0 ] = (1/ (xln2)) [ 1 + 1 / lnx ]